The result REF  follows from using chain rules, product rules and the formula of change of variables:
\(\int _{\mathbb {R}^n}|(\chi f)(\Phi (x))|^pdx=\int _{\mathbb {R}^n}|\chi (y) f(y)|^p|\det \nabla (\Phi ^{-1})(y)|dy,\quad 0<p<\infty ,\quad f\in L^p_\mathrm {loc}(\mathbb {R}^n).\) 
Notice that \(|\det \nabla (\Phi ^{-1})|\)  is a bounded smooth function, and all nonzero derivatives of \(\Phi \)  and \(\chi \)  are bounded. We leave the details to readers.
For REF , see for example [1]} and [2]}.
